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User blog:P進大好きbot/Historical Background of the Ill-definedness of UNOCF
Since I am repeating the explanations about issues on UNOCF so many times, I summarise the historical background of the ill-definedness of UNOCF. For a brief explanation, see this blog post. = Original Definition = UNOCF is originally introduced by Username5243 here. Although UNOCF is intended to form an OCF, the definition has many issues, which make UNOCF ill-defined. Uncountable Ordinal UNOCF uses an unspecified function \(f\) in the definition of \(\psi(\alpha)\) for an ordinal \(\alpha\) of cofinality \(\Omega\). The function \(f\), which depends on \(\alpha\) because it is required to satisfy \(f(\Omega) = \alpha\), is not unique at all, and the result of \(\psi(\alpha)\) heavily depends on the choice of \(f\). Even if we are allowed to apply any choice of \(f\), the resulting \(\psi\) can behave awfully different from usual OCFs because \(f\) is not required to be strictly increasing or continuous. Moreover, there is no known explicit method to define such an \(f\) for any ordinals \(\alpha\) of cofinality \(\Omega\). Moreover, UNOCF is supposed to satisfy weird equalities \begin{eqnarray*} \psi(\Omega_2+\Omega) = \psi(\varepsilon_{\Omega+1}+1) \end{eqnarray*} and \begin{eqnarray*} \psi(\Omega_2^2) = \psi(\zeta_{\Omega+1}). \end{eqnarray*} At least, those do not hold for Buchholz's OCF, and hence it implies that it is impossible to formalise UNOCF as an addition-based OCF while googologists who believe that UNOCF works often state that UNOCF is an addition-based OCF which coincides with Buchholz's OCF up to its limit. (I received an opinion that the \(\psi\)'s in the right hand sides are not UNOCF, but there seems no such a description in the original explanation. In mathematics, the same symbol in a single equality means the same object unless specified, and hence this opinion is not reasonable.) Stage Cardinal UNOCF uses the stage cardinal \(T\), which is ill-defined by the reason explained here. It is intended to diagonalise the sequence "\((I_{\alpha},M_{\alpha},K_{\alpha},\ldots)\)" of large cardinal axioms, but there is no known reasonable sequence of large cardinal axioms whose first three entries are \(I_{\alpha}\), \(M_{\alpha}\), and \(K_{\alpha}\). Namely, what Username5243 did is just showing the first three entries and behaving as if they were portions of some existing sequence. Guessed Expansions Since UNOCF is not fully defined, many googologists tried to guess how it is supposed to work. However, there is also a problem. I recall that they often state that the benefit of UNOCF is that it is easier to understand than usual OCFs. Actually, few googologists understand usual OCFs based on higher inaccessibility, because those definitions are difficult. Therefore the guessed expansions are mainly given by googologists who do not understand the definitions of usual OCFs. As a result, expressions in UNOCF are guessed to expand in an awfully different way from those in usual OCFs. Although it is irrelevant to the original definition of UNOCF itself, the guessed unreasonable expansions have been regarded as "official" expansions for them. The situation is quite mysterious, but it is evidently due to the lack of the formality in the original definition of UNOCF. For example, what are the worst "official" expansions are \begin{eqnarray*} \psi(\varepsilon_0+1) = \psi(\varepsilon_0) + \psi(\varepsilon_0) + \psi(\varepsilon_0) + \cdots \end{eqnarray*} and \begin{eqnarray*} \psi(\psi_3(0)) = \psi(\psi_1(\psi_2(\cdots \psi_2(0) \cdots))), \end{eqnarray*} where \(\psi_1\), \(\psi_2\), and \(\psi_3\) are shorthands of \(\psi_{\Omega}\), \(\psi_{\Omega_2}\), and \(\psi_{\Omega_3}\) respectively. They require UNOCF to be completely different from usual OCFs, although googologists who do not understand usual OCFs often state "It coincides with Buchholz's OCF in this realm". Needless to say, Buchholz's OCF does not satisfy such weird properties. Then why do they believe that UNOCF in this realm coincides with Buchholz's OCF? It is simply because they do not know usual OCFs such as Buchholz's OCF. Actually, they sometimes state that those expansions are valid for Buchholz's OCF "because" it is the same as UNOCF. Right, this is a typical circular logic. "Why do expressions in UNOCF expand in those weird ways?" "Simply because UNOCF up tp \(\psi(\Omega_{\omega})\) is the same as Buchholz's OCF. Why couldn't you understand such a trivial thing?" "You mean, expressions in Buchholz's OCF expand in those ways, right?" "Why not? They expand in these ways, because Buchholz's OCF is the same as UNOCF! I WIN!!!!" = Alternative Definition = Since there are many issues in the original definition, several googologists are trying to formalise it. As an OCF There are no justification of UNOCF as an actual OCF, because it admits weird equalities such as \(\psi(\Omega_2+\Omega) = \psi(\varepsilon_{\Omega+1}+1)\) and weird expansions such as \(\psi(\varepsilon_0+1) = \psi(\varepsilon_0) + \psi(\varepsilon_0) + \psi(\varepsilon_0) + \cdots\) which are unfortunately regarded as "official" expansions of UNOCF by googologists who believe that UNOCF can work. Why do they insist such expansions, which prevent a reasonable formalisation? Simply because such expansions are believed to be easier to understand than those of usual OCFs. The problem is that googologists who believe that UNOCF were the greatest OCF do not know usual OCFs well, and hence do not know a way to define an actual OCF. At least, in order to define \(\psi(\Omega)\), we need to define \(\psi(\alpha)\) for any countable ordinal \(\alpha\), if we require \(\psi\) to be an actual OCF. On the other hand, there is no agreed-upon definition of \(\psi(\omega_1^{\textrm{CK}})\) in UNOCF such that it fits the "guessed" expansion \(\psi(\varepsilon_0+1) = \psi(\varepsilon_0) + \psi(\varepsilon_0) + \psi(\varepsilon_0) + \cdots\). As a Notation There are many trials to formalise UNOCF as a notation, which can be defined in arithmetic. However, in order to defined an expansion of an expressions of the form \(\psi(\psi_{\Omega_{a+1}}(\Omega_{b+1}))\), we usually need to define a binary relation \(a < b\). Therefore when we deal with an OCF-like recursive notation above \(\psi_0(\Omega_{\omega})\) with respect to Buchholz's OCF, we need to equip it with an explicit algorithm to determine whether \(a < b\) holds or not. There is a deeply related notion called an ordinal notation. Since the definition of an ordinal notation is widely misunderstood in this community, please see this blog post if you do not know the precise definition. In order to create an ordinal notation, we usually need an actual OCF or much more complicated mathematical theories. I recall that they are trying to formalise UNOCF into an arithmetic notation because it is really difficult to formalise it into an actual OCF. Therefore the usual strategy to create an ordinal notation is useless for this purpose. The only well-defined stuff which formalises UNOCF is NIECF, which is created by Nayuta Ito here currently up to what is denoted by \(\psi(I)\) in UNOCF. How about other attempts? They are ill-defined by many reasons. For example, many of them lack the declaration of the set of expressions, include functions without declaration of domains, use an overloaded equality, rely on unwritten rule sets based on pattern matching, and so on. Right, there are plenty of problems. At least, I have never seen an example other than NIECF such that an algorithm for \(<\) is completely defined. It is just kept WIP, or omitted as if it were trivial. I have seen a circular logic on \(<\) so many times. "Your rule sets of expansions includes \(<\). What is the definition of \(<\)?" "Obviously, it is the comparison of ordinals. Why couldn't you understand such a trivial thing?" "But you are formalising UNOCF into a notation, in which no ordinals are allowed to involved." "It precisely means the comparison of the ordinals corresponding to expressions. Why couldn't you understand such a trivial thing?" "What does the ordinal corresponding to an expression in your notation mean?" "It is the strength. More precisely, an expression \(s\) corresponds to an ordinal \(\alpha\) if the FGH at \(s\) is approximated by the FGH at \(\alpha\). Why couldn't you understand such a trivial thing?" "But in order to define the FGH, you need to define the full rule set of expansions. Referring to the FGH in the definition of expansions is a circular logic." "In fact, we do not need the FGH. I just used the FGH in order to make you understand better. For example, define a map \(o\) which assigns an expressing \(s\) to an ordinal \(o(s)\) in the following recursive way: If \(s = 0\), then \(o(s) = 0\). If \(s\) is the successor of \(t\), then \(o(s) = o(t) + 1\). When \(s\) is a limit, then \(o(s)\) is the limit of \(o(sn)\) on \(n \in \mathbb{N}\). Quite obvious. Why couldn't you understand such a trivial thing?" "But the definition of \(o(s)\) includes \(sn\), which should be defined by the rule set for expansions. Refering to \(o\) in the definition of expansions is a circular logic." "Hey, stop it. Why should I help a beginner to understand such a trivial thing? Since UNOCF works, expressions in UNOCF corresponds to ordinals! I WIN!!!!" = Does It Work? = Several googologists who have heard of the ill-definedness of UNOCF still use it by stating that there is an agreed-upon structure of UNOCF even if the full definition has not been formalised. However, this statement also includes problems. As an OCF Needless to say, the stage cardinal \(T\) is ill-defined. Therefore we do not have to point out the lack of the full definition of \(\psi\) itself. We do not have an agreed-upon structure of UNOCF including \(T\). As a Notation What does an agreed-upon expansion? It is reasonable to assume that when we have an agreed-upon expansion of an expression \(s\), the result of the computation of \(s2\) should be independent of a googologist who computes it. Namely, the expression \(s2\) should be unique. What does "uniqueness" mean in this context? If we have no definition of an overloaded equality, then it should be the uniqueness as formal strings. However, googologists output different results of \(s2\) when \(s\) is a sufficiently complicated expression such as \(\psi(\varphi_{\omega}(1))\), \(\psi(\varphi_{\Omega}(1))\), and \(\psi(\Phi_I(1))\). Then do we have a well-defined overloaded equality? In order to define a relation such as an overloaded equality, we need to define the domain as an explicit set of formal strings. However, I sometimes come across a circular logic here. "I have defined the set \(S\) of formal strings and the equality \(s = t\) on \((s,t) \in S^2\)." "Then what does the equality \(\psi(\varphi_2(0)) = \psi(\zeta_0)\) mean?" "As I said, I have defined the equality. Why couldn't you understand such a trivial thing?" "But neighther \(\psi(\varphi_2(0))\) nor \(\psi(\zeta_0)\) belongs to \(S\)." "It does not matter, because they are equivalent to an expression in \(S\). Why couldn't you understand such a trivial thing?" "Then what does the equivalence precisely mean?" "As I said, I have defined the equality. The equivalence is just given by the equality. Why couldn't you understand such a trivial thing?" "But they are outsude the domain of the equality, and hence referring to the equality in the definition of the equivalence of them with another expression in \(S\) is a circular logic." "To be more precise, the equivalence is given by the equality of the corresponding ordinals. Why couldn't you understand such a trivial thing?" "But in order to define the corresponding ordinals, you need to define the expansions for all expressions which are necessary to define the FGH or \(o\). For this purpose, you need to define the full rule set of expansions. For this purpose, you need to define an algorithm to compute \(<\). However, defining \(<\) is usually much more difficult than defining \(=\), because we can define \(a = b\) as the negation of \((a < b) \lor (b < a)\)." "Hey, stop it. Why should I help a beginner to understand such a trivial thing? Since UNOCF works for experts, we have agreed-upon expansions! I WIN!!!!" Category:Blog posts